b) Prove by mathematical induction that n'-n is divisible by 3 for all n>1.
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- Ouestion3. Prove the following inequality by induction n< 2", n e Z+7. For n 2 1, in how many out of the n! permutations T = (T(1), 7(2),..., 7 (n)) of the numbers {1, 2, ..., n} the value of 7(i) is either i – 1, or i, or i +1 for all 1 < i < n? Example: The permutation (21354) follows the rules while the permutation (21534) does not because 7(3) = 5. Hint: Find the answer for small n by checking all the permutations and then find the recursive formula depending on the possible values for 1(n).3. Prove that if n is an integer leaving remainder 1 after division by 4, then = n. Provide the type of your proof and all details.
- . Prove that n2 + 1 2n, where n is a positive integer with 1 n 4.Use two ways to count the number of r-combinations of[n]={1,2,…,n}that contains 1 or 2 or 3. First, separate cases with Case 1 counting all r-combinations of [n] that contains 1 , Case 2 counting all r-combination of [n] that contains 2 but not 1 , and Case 3 counting allrcombination of [n] that contains 3 but not 1 or 2 . Second, count all r-combinations of [n] not containing any of1,2,3and use the subtraction rule.Use induction to show that 4n < 2nfor all n ≥5